Polyacrylamide (PAA)

Polyacrylamide is a water-soluble polymer and was selected because 0.05% PAA in

water had previously been used by Poole et al. _{(2005) during the first observations of}

the ‘cat’s ears’ effect. PAA is transparent making it ideal for obtaining laser Doppler anemometry measurements (see Chapter 3 for details). It is a viscoelastic shear- thinning fluid and generally thought of as having a ‘very flexible’ molecular structure (Walters et al. _{(1990)). This flexibility means that the fluid has more}

pronounced elastic properties than other water-soluble polymers such as xanthan gum or carboxymethylcellulose. The polyacrylamide used during the investigation

was Separan AP 273 E with a molecular weight of approximately 2x106g/mol.

Figure 2.6 shows the variation of shear viscosity with shear rate for 16 concentrations of PAA in water varying between 25ppm (i.e. 0.0025% w/w) and 0.35%. It can clearly be seen that even at low concentrations PAA is a weakly shear- thinning fluid. The Carreau-Yasuda model (Equation 2.12) has been fitted to each set of results and the corresponding curve is also shown in Figure 2.6. The filled symbols indicate those results deemed to be below the effective resolution of the

Fluid Characterisation

rheometer or adversely affected by secondary flows and hence not included in the fit. Table 2.1 presents the corresponding fitting parameters for the fits shown in Figure 2.6. From the Carreau-Yasuda fit a zero shear rate viscosity can be estimated for each concentration of PAA, these values allow the determination of the critical overlap concentration.

Figure 2.7 shows the oscillatory shear data for concentrations above 0.05% PAA, the results for concentrations below 0.05% were deemed to be too close to the limits of the rheometer and hence are not presented. Figure 2.7 (a) shows the variation in the storage modulus with angular frequency and Figure 2.7 (b) shows the variation in the loss modulus; from this data it is possible to calculate the dynamic viscosity and the dynamic rigidity as described earlier in the chapter and these values can be seen in Figures 2.7 (c) and 2.7 (d). Figure 2.7 (c) shows the dynamic rigidity for the various

concentrations, calculated from G′, which should be zero for an inelastic fluid. As

can be seen from Figures 2.7 (a) and (c) neither G′ nor the dynamic rigidity are zero

hence the fluids are elastic. G′and G′′ both increase as the concentration of polyacrylamide increases, hence the dynamic viscosity and dynamic rigidity are also seen to increase with an increase in concentration.

Figure 2.8 shows the variation in zero shear rate viscosity with concentration. Two clear power-law ranges can be seen indicating the dilute range and the semi-dilute range. The point at which the curves intersect allows the determination of the critical

overlap concentration, c*_{, here found to be approximately 0.03% PAA. As}

mentioned earlier in the chapter the slope of the dilute curve should be around 1 and Figure 2.8 shows that for this polymer it is, in fact, 0.78. The slope of the semi-dilute power law curve is 3.3 confirming that polyacrylamide is a flexible polymer. The filled symbols show the concentrations selected for the detailed fluid dynamic investigation. The concentrations of PAA chosen for these investigations were

0.05% (c/c*_{=1.67) to correspond to the measurements performed by Poole} et al_.

(2005), 0.03% because this concentration was determined to be approximately equal

to c*_{, 0.01% (}c/c*_{=0.33), which is well within the dilute range and 0.3% (}c/c*₌₁₀₎

was selected with the intention of decreasing the Reynolds number in an attempt to minimise the effects of inertia. Extensional rheology measurements were performed

Fluid Characterisation

on these fluids in order to determine a relaxation time for the fluid, which could be

used to estimate a Deborah number, De_C.

Because 0.01% PAA is in the dilute range neither N_{1, oscillatory nor extensional}

measurements were obtainable. The shear viscosity variation with shear rate for this fluid can be seen in Figure 2.6 (b). Figure 2.9 shows the material properties for 0.03% PAA. It is seen to be shear thinning and, although the oscillatory data for 0.03% PAA was deemed to be too close to the limits to be reliable, the dynamic

viscosity has been included in this figure. At this concentration the G′′ data makes

up approximately 99% (Barnes et al.) of the measurement while the G′ data is the

remaining 1% meaning the G′′ data is still effectively reliable. The dynamic

viscosity, which corresponds to the G′′ data, agrees well with the zero shear rate viscosity as the shear rate tends towards zero. Figure 2.10 shows the material properties for 0.05% PAA and Figure 2.11 the material properties for 0.3% PAA. Both fluids are seen to be shear thinning and elastic with the dynamic viscosity tending towards the same value as the zero shear rate viscosity as the shear rate tends towards zero as expected (Equation 2.8).

Figure 2.12 presents the normal force data for the 0.03%, 0.05% and 0.3% PAA, along with the corresponding relaxation times (estimated using N₁ =2

λτγ

discussed in Chapter 1). For each concentration the first normal-stress difference is observed to increase with an increase in the shear rate and the relaxation time is seen to decrease with an increase in the shear rate. As might be expected at a given shear

rate, the higher the concentration, the larger N_{1 and the relaxation time. The power}

law index of the first-normal stress difference (on a log-log plot at shear rate ranges between approximately 700 and 6000) is approximately the same for each of the three concentrations of polyacrylamide with a value of 0.8 as shown by the full black lines. This result implies that the polymer itself determines this slope, rather than the concentration.

Figure 2.13 shows the extensional rheology data for two samples of 0.03% PAA. The filament diameter is seen to decay over time as expected and is fully broken up after approximately 0.1s. The formula given in equation 2.14 was fitted to the data

Fluid Characterisation

(shown as a full line in Figure 2.13) and a relaxation time of λ=0.025s was estimated

from this fit. The fitting parameters are given in Table 2.2. Figure 2.14 is a selection of photographs taken with a high-speed camera during the tests. It is clear from these images that the initial sample is correctly loaded and the filament diameter decay over time can be seen. Figure 2.15 gives the extensional rheology measurements for two samples of 0.05% PAA, again the filament diameter is seen to decay over time but for the 0.05% PAA solution the break-up time is longer than for the 0.03% solution, indicating that 0.05% PAA has a longer relaxation time than 0.03% PAA. As for the 0.03% PAA, the data was fitted to Equation 2.14 and a relaxation time of

λ=0.056s was estimated. Figure 2.16 shows some of the high-speed camera images

taken during the 0.05% PAA measurements. Clearly the sample has been correctly loaded and a suitable cylindrical filament has formed during the test. Figure 2.17 shows the experimental data for two samples of 0.3% PAA. As might have been expected the 0.3% PAA solution takes much longer to break up than either the 0.03% or 0.05% solutions. In this case the relaxation time was estimated to be

λ=3.44s, which is an order of magnitude larger than that for the lower concentrations.

Figure 2.18 is a sample of the high-speed images obtained during a test on 0.3% PAA. The images show that the fluid sample was loaded correctly and that the filament is cylindrical throughout its decay. On closer inspection of the high-speed camera images (Figures 2.14, 2.16 and 2.18) the filament in each case was found to be perfectly cylindrical with no discernible ‘bowing’, which means that the flow at the filament mid-point is purely extensional and provides a level of confidence in any relaxation times obtained from the fitted data.